3.21.47 \(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [2047]

Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}} \]

[Out]

32/3003*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^4/d^4/(e*x+d)^(7/2)+16/429*(-a*e^2+c*d^2)^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)^(5/2)+12/143*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(7/2)/c^2/d^2/(e*x+d)^(3/2)+2/13*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+d)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(32*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*c^4*d^4*(d + e*x)^(7/2)) + (16*(c*d
^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (12*(c*d^2 - a*e^
2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*c^2*d^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*
x + c*d*e*x^2)^(7/2))/(13*c*d*Sqrt[d + e*x])

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (6 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx}{13 d}\\ &=\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (24 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{143 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{429 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 142, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (143 d^2+182 d e x+63 e^2 x^2\right )+c^3 d^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(13*d + 7*e*x) - 2*a*c^2*d^2*e^2
*(143*d^2 + 182*d*e*x + 63*e^2*x^2) + c^3*d^3*(429*d^3 + 1001*d^2*e*x + 819*d*e^2*x^2 + 231*e^3*x^3)))/(3003*c
^4*d^4*Sqrt[d + e*x])

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Maple [A]
time = 0.82, size = 160, normalized size = 0.69

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (-231 c^{3} d^{3} e^{3} x^{3}+126 a \,c^{2} d^{2} e^{4} x^{2}-819 c^{3} d^{4} e^{2} x^{2}-56 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -1001 c^{3} d^{5} e x +16 e^{6} a^{3}-104 e^{4} d^{2} a^{2} c +286 d^{4} e^{2} c^{2} a -429 d^{6} c^{3}\right )}{3003 \sqrt {e x +d}\, c^{4} d^{4}}\)	\(160\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-231 c^{3} d^{3} e^{3} x^{3}+126 a \,c^{2} d^{2} e^{4} x^{2}-819 c^{3} d^{4} e^{2} x^{2}-56 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -1001 c^{3} d^{5} e x +16 e^{6} a^{3}-104 e^{4} d^{2} a^{2} c +286 d^{4} e^{2} c^{2} a -429 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 c^{4} d^{4} \left (e x +d \right )^{\frac {5}{2}}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3003/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(c*d*x+a*e)^3*(-231*c^3*d^3*e^3*x^3+126*a*c^2*d^2*e^4*x^2-81
9*c^3*d^4*e^2*x^2-56*a^2*c*d*e^5*x+364*a*c^2*d^3*e^3*x-1001*c^3*d^5*e*x+16*a^3*e^6-104*a^2*c*d^2*e^4+286*a*c^2
*d^4*e^2-429*c^3*d^6)/c^4/d^4

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Maxima [A]
time = 0.31, size = 321, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (231 \, c^{6} d^{6} x^{6} e^{3} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \, {\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (x e + d\right )}}{3003 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*c^6*d^6*x^6*e^3 + 429*a^3*c^3*d^6*e^3 - 286*a^4*c^2*d^4*e^5 + 104*a^5*c*d^2*e^7 - 16*a^6*e^9 + 63*
(13*c^6*d^7*e^2 + 9*a*c^5*d^5*e^4)*x^5 + 7*(143*c^6*d^8*e + 299*a*c^5*d^6*e^3 + 53*a^2*c^4*d^4*e^5)*x^4 + (429
*c^6*d^9 + 2717*a*c^5*d^7*e^2 + 1469*a^2*c^4*d^5*e^4 + 5*a^3*c^3*d^3*e^6)*x^3 + 3*(429*a*c^5*d^8*e + 715*a^2*c
^4*d^6*e^3 + 13*a^3*c^3*d^4*e^5 - 2*a^4*c^2*d^2*e^7)*x^2 + (1287*a^2*c^4*d^7*e^2 + 143*a^3*c^3*d^5*e^4 - 52*a^
4*c^2*d^3*e^6 + 8*a^5*c*d*e^8)*x)*sqrt(c*d*x + a*e)*(x*e + d)/(c^4*d^4*x*e + c^4*d^5)

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Fricas [A]
time = 3.52, size = 349, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (429 \, c^{6} d^{9} x^{3} + 8 \, a^{5} c d x e^{8} - 16 \, a^{6} e^{9} - 2 \, {\left (3 \, a^{4} c^{2} d^{2} x^{2} - 52 \, a^{5} c d^{2}\right )} e^{7} + {\left (5 \, a^{3} c^{3} d^{3} x^{3} - 52 \, a^{4} c^{2} d^{3} x\right )} e^{6} + {\left (371 \, a^{2} c^{4} d^{4} x^{4} + 39 \, a^{3} c^{3} d^{4} x^{2} - 286 \, a^{4} c^{2} d^{4}\right )} e^{5} + {\left (567 \, a c^{5} d^{5} x^{5} + 1469 \, a^{2} c^{4} d^{5} x^{3} + 143 \, a^{3} c^{3} d^{5} x\right )} e^{4} + {\left (231 \, c^{6} d^{6} x^{6} + 2093 \, a c^{5} d^{6} x^{4} + 2145 \, a^{2} c^{4} d^{6} x^{2} + 429 \, a^{3} c^{3} d^{6}\right )} e^{3} + 13 \, {\left (63 \, c^{6} d^{7} x^{5} + 209 \, a c^{5} d^{7} x^{3} + 99 \, a^{2} c^{4} d^{7} x\right )} e^{2} + 143 \, {\left (7 \, c^{6} d^{8} x^{4} + 9 \, a c^{5} d^{8} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3003 \, {\left (c^{4} d^{4} x e + c^{4} d^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

2/3003*(429*c^6*d^9*x^3 + 8*a^5*c*d*x*e^8 - 16*a^6*e^9 - 2*(3*a^4*c^2*d^2*x^2 - 52*a^5*c*d^2)*e^7 + (5*a^3*c^3
*d^3*x^3 - 52*a^4*c^2*d^3*x)*e^6 + (371*a^2*c^4*d^4*x^4 + 39*a^3*c^3*d^4*x^2 - 286*a^4*c^2*d^4)*e^5 + (567*a*c
^5*d^5*x^5 + 1469*a^2*c^4*d^5*x^3 + 143*a^3*c^3*d^5*x)*e^4 + (231*c^6*d^6*x^6 + 2093*a*c^5*d^6*x^4 + 2145*a^2*
c^4*d^6*x^2 + 429*a^3*c^3*d^6)*e^3 + 13*(63*c^6*d^7*x^5 + 209*a*c^5*d^7*x^3 + 99*a^2*c^4*d^7*x)*e^2 + 143*(7*c
^6*d^8*x^4 + 9*a*c^5*d^8*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^4*d^4*x*e + c^4*
d^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(5/2)*sqrt(d + e*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2990 vs. \(2 (214) = 428\).
time = 1.44, size = 2990, normalized size = 12.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

2/45045*(429*c^2*d^5*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*
d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d
^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2
*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^(-1) - 6006*a*c*d^4*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3
- 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c
*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-1) - 429*c^2*d^4*((35*sqrt(-c*d
^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*
sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)
*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)) + 1501
5*a^2*d^3*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) + (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^
2*e + a*e^3)*a*e^2)/(c*d))*e + 2574*a*c*d^3*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c
^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35
*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*(
(x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e + 39*c^2*d^3*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^
10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e +
a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)
/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^
3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d)*c*d*e - c*d^2*e +
a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e - 858*a*c*d^2*((35*sq
rt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e
^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x
*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))
*e^2 - 5*c^2*d^2*((693*sqrt(-c*d^2*e + a*e^3)*c^6*d^12 - 63*sqrt(-c*d^2*e + a*e^3)*a*c^5*d^10*e^2 - 70*sqrt(-c
*d^2*e + a*e^3)*a^2*c^4*d^8*e^4 - 80*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^6*e^6 - 96*sqrt(-c*d^2*e + a*e^3)*a^4*c^
2*d^4*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*c*d^2*e^10 - 256*sqrt(-c*d^2*e + a*e^3)*a^6*e^12)*e^(-5)/(c^6*d^6)
+ (3003*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^5*e^15 - 9009*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^
4*e^12 + 12870*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^3*e^9 - 10010*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
9/2)*a^2*e^6 + 4095*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*a*e^3 - 693*((x*e + d)*c*d*e - c*d^2*e + a*e^3)
^(13/2))*e^(-11)/(c^6*d^6))*e^2 - 9009*a^2*d^2*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e +
d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e
^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e + 1287*a^2*d*((15*sqrt(-c*d^2*e + a*e^3)*c^3*
d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^
3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c
*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^3 + 26*a*c*d*(
(315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2
*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(
-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772
*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 -
1540*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/
(c^5*d^5))*e^3 - 143*a^2*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt
(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^
8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e
+ a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e +
a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e^4)*e^(-1)

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Mupad [B]
time = 1.40, size = 383, normalized size = 1.64 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^4\,\sqrt {d+e\,x}\,\left (\frac {106\,a^2\,e^4}{429}+\frac {46\,a\,c\,d^2\,e^2}{33}+\frac {2\,c^2\,d^4}{3}\right )-\frac {\sqrt {d+e\,x}\,\left (32\,a^6\,e^9-208\,a^5\,c\,d^2\,e^7+572\,a^4\,c^2\,d^4\,e^5-858\,a^3\,c^3\,d^6\,e^3\right )}{3003\,c^4\,d^4\,e}+\frac {2\,c^2\,d^2\,e^2\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^3\,\sqrt {d+e\,x}\,\left (10\,a^3\,c^3\,d^3\,e^6+2938\,a^2\,c^4\,d^5\,e^4+5434\,a\,c^5\,d^7\,e^2+858\,c^6\,d^9\right )}{3003\,c^4\,d^4\,e}+\frac {6\,c\,d\,e\,x^5\,\left (13\,c\,d^2+9\,a\,e^2\right )\,\sqrt {d+e\,x}}{143}+\frac {2\,a\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4+715\,a\,c^2\,d^4\,e^2+429\,c^3\,d^6\right )}{1001\,c^2\,d^2}+\frac {2\,a^2\,e\,x\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-52\,a^2\,c\,d^2\,e^4+143\,a\,c^2\,d^4\,e^2+1287\,c^3\,d^6\right )}{3003\,c^3\,d^3}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x^4*(d + e*x)^(1/2)*((106*a^2*e^4)/429 + (2*c^2*d^4)/3 + (46*a
*c*d^2*e^2)/33) - ((d + e*x)^(1/2)*(32*a^6*e^9 - 208*a^5*c*d^2*e^7 - 858*a^3*c^3*d^6*e^3 + 572*a^4*c^2*d^4*e^5
))/(3003*c^4*d^4*e) + (2*c^2*d^2*e^2*x^6*(d + e*x)^(1/2))/13 + (x^3*(d + e*x)^(1/2)*(858*c^6*d^9 + 5434*a*c^5*
d^7*e^2 + 2938*a^2*c^4*d^5*e^4 + 10*a^3*c^3*d^3*e^6))/(3003*c^4*d^4*e) + (6*c*d*e*x^5*(9*a*e^2 + 13*c*d^2)*(d
+ e*x)^(1/2))/143 + (2*a*x^2*(d + e*x)^(1/2)*(429*c^3*d^6 - 2*a^3*e^6 + 715*a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4))
/(1001*c^2*d^2) + (2*a^2*e*x*(d + e*x)^(1/2)*(8*a^3*e^6 + 1287*c^3*d^6 + 143*a*c^2*d^4*e^2 - 52*a^2*c*d^2*e^4)
)/(3003*c^3*d^3)))/(x + d/e)

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